Let \(f(t)\) be a piecewise continuous function on \([0,\infty )\). The Laplace transform of \(f\) is the function \(F(s)=\displaystyle \int _0^{\infty } f(t)e^{-st}\,dt\), on the set of \(s\) values for which the integral converges.

(a)
By direct calculation, determine the Laplace transform of the function graphed below (assume the function continues to be equal to zero to the right of \(x=4\)) for \(s>0\). Put the answer in the box provided.

The graph of a piecewise continuous function. [Picture]
Figure 1: Graph of a piecewise continuous function
(b)
Fill-in the missing details (after each of the ellipses) to show by direct calculation that the Laplace transform of \(f(t)=\cos {(\omega t)}\) where \(\omega >0\), is \(\displaystyle \frac {s}{s^2+\omega ^2}\) for \(s>0\).

If \(s=0\) then \(\displaystyle \int _0^{\infty } \cos {(\omega t)}e^{-st}\,dt=\int _0^{\infty } \cos {(\omega t)}\,dt\), which ... . If

\(s<0\) then \(\displaystyle \int _0^{\infty } \cos {(\omega t)}e^{-st}\,dt\) also ... . For \(s> 0\), let

\(\displaystyle A=\int _0^{\infty } \cos {(\omega t)}e^{-st}\,dt\). By ... with \(u=\cos {(\omega t)}\)

and ... we get that

\(A=\) ...       \(\displaystyle \left .\:^{\;}\right |_0^{\infty } - \frac {\omega }{s} \int _0^{\infty }\) ...      \(dt\).

For \(s>0\), \(\displaystyle \lim _{t\rightarrow \infty } -\frac {\cos {(\omega t)}}{s}e^{-st}=\) ...      

so \(\displaystyle A=\frac {1}{s} \; -\) ...      .

By ... , with \(\displaystyle u=\sin {(\omega t)}\) and ... we get

\(\displaystyle A=\frac {1}{s} \; - \; \frac {\omega }{s} [\) ...      \(]\).

For \(s>0\), \(\displaystyle \lim _{t\rightarrow \infty } -\frac {\sin {(\omega t)}}{s}e^{-st}=\) ...      .

Then \(A=\frac {1}{s} \; - [\) ...      \(]A\). Solving this for \(A\)

we get \(A=\) , Hence the ... of \(\cos (\omega t)\) is

\(A=\) ... for \(s>0\).

(c)
One can also show that \(\displaystyle \Laplace {\sin (\omega t))}=\frac {\omega }{s^2+\omega ^2}\) for \(s>0\). Use the First Shifting Property to derive expressions for the Laplace transforms \(\Laplace {e^{at}\cos {(\omega t)}}\) and \(\Laplace {e^{at}\sin {(\omega t)}}\) for \(s>a\), where \(a,\omega >0\) are constants. Put the answers in the box provided.

(d)
Using \(\Laplace {y^{\,\prime }}=s\Laplace {y}-y(0)\), find the Laplace transform of the solution to the IVP \(y^{\,\prime }+y=e^{-t}\sin {(2t)}\), \(y(0)=1\) by taking the Laplace transform of both sides of the differential equation and then solving for \(\Laplace {y}\) algebraically. Put the answer in the box provided.